Shearing and Translation in Pythagorean Pants

The most famous of all diagrams depicting Euclid's I.47 is known under many names, Bride's Chair being probably the most popular. In Russia, however, the common term is the Pythagorean Pants. There is even a short verse claiming that they are equal in all directions - whatever this may mean.

Unfolding the diagram in Proof 69 brings up the Euclidean Bride's Chair but without the accompanying cradle of interior lines. So I judged that to mention the Pythagorean Pants would be quite appropriate alongside the simplified drawing. Now, if one wonders why Euclid did not see the simple Proof 69, then the unfolded diagram which is even closer to the drawing implied by his proof of I.47 adds to the mystery.